Properties

Label 1617.31
Modulus $1617$
Conductor $77$
Order $30$
Real no
Primitive no
Minimal no
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1617, base_ring=CyclotomicField(30))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,5,18]))
 
pari: [g,chi] = znchar(Mod(31,1617))
 

Basic properties

Modulus: \(1617\)
Conductor: \(77\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(30\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{77}(31,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1617.bm

\(\chi_{1617}(31,\cdot)\) \(\chi_{1617}(313,\cdot)\) \(\chi_{1617}(460,\cdot)\) \(\chi_{1617}(619,\cdot)\) \(\chi_{1617}(1048,\cdot)\) \(\chi_{1617}(1060,\cdot)\) \(\chi_{1617}(1489,\cdot)\) \(\chi_{1617}(1501,\cdot)\)

sage: chi.galois_orbit()
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q(\zeta_{15})\)
Fixed field: 30.0.13209167403604364499542354001933559191813355687.1

Values on generators

\((1079,199,442)\) → \((1,e\left(\frac{1}{6}\right),e\left(\frac{3}{5}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(13\)\(16\)\(17\)\(19\)\(20\)
\( \chi_{ 1617 }(31, a) \) \(-1\)\(1\)\(e\left(\frac{14}{15}\right)\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{7}{30}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{17}{30}\right)\)\(e\left(\frac{19}{30}\right)\)\(e\left(\frac{1}{10}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1617 }(31,a) \;\) at \(\;a = \) e.g. 2